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<p class=MsoNormal><font size=3 face="Times New Roman"><span style='font-size:
12.0pt'>Hi<o:p></o:p></span></font></p>
<p class=MsoNormal><font size=3 face="Times New Roman"><span style='font-size:
12.0pt'><o:p> </o:p></span></font></p>
<p class=MsoNormal><font size=3 face="Times New Roman"><span style='font-size:
12.0pt'>The shape of an elliptical orbit depends on the energy and angular
momentum of the satellite. <o:p></o:p></span></font></p>
<p class=MsoNormal><font size=3 face="Times New Roman"><span style='font-size:
12.0pt'><o:p> </o:p></span></font></p>
<p class=MsoNormal><font size=3 face="Times New Roman"><span style='font-size:
12.0pt'>A mathematical analysis reveals that the ellipses two semi axes are
given by <o:p></o:p></span></font></p>
<p class=MsoNormal><font size=3 face="Times New Roman"><span style='font-size:
12.0pt'><span style='position:relative;top:13.0pt;mso-text-raise:-13.0pt'><!--[if gte vml 1]><v:shapetype
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</xml><![endif]--> for a bound satellite.<o:p></o:p></span></font></p>
<p class=MsoNormal><font size=3 face="Times New Roman"><span style='font-size:
12.0pt'><o:p> </o:p></span></font></p>
<p class=MsoNormal><font size=3 face="Times New Roman"><span style='font-size:
12.0pt'>The special case of a circular orbit dictates certain requirements on <span
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</xml><![endif]--> and this is not as easy to obtain as an elliptical
orbit. A demonstration is possible with a ‘conical pendulum’ made
to orbit under a central force. <o:p></o:p></span></font></p>
<p class=MsoNormal><font size=3 face="Times New Roman"><span style='font-size:
12.0pt'><o:p> </o:p></span></font></p>
<p class=MsoNormal><font size=3 face="Times New Roman"><span style='font-size:
12.0pt'>Keith Irvine<o:p></o:p></span></font></p>
<p class=MsoNormal><font size=3 face="Times New Roman"><span style='font-size:
12.0pt'>AIC, <st1:place w:st="on"><st1:City w:st="on">Auckland</st1:City></st1:place><o:p></o:p></span></font></p>
<p class=MsoNormal><font size=3 face="Times New Roman"><span style='font-size:
12.0pt'><o:p> </o:p></span></font></p>
<p class=MsoNormal><font size=2 color=navy face=Arial><span style='font-size:
10.0pt;font-family:Arial;color:navy'><o:p> </o:p></span></font></p>
<p class=MsoNormal><font size=2 color=navy face=Arial><span style='font-size:
10.0pt;font-family:Arial;color:navy'><o:p> </o:p></span></font></p>
<div>
<div class=MsoNormal align=center style='text-align:center'><font size=3
face="Times New Roman"><span style='font-size:12.0pt'>
<hr size=2 width="100%" align=center tabindex=-1>
</span></font></div>
<p class=MsoNormal><b><font size=2 face=Tahoma><span style='font-size:10.0pt;
font-family:Tahoma;font-weight:bold'>From:</span></font></b><font size=2
face=Tahoma><span style='font-size:10.0pt;font-family:Tahoma'>
phys-teach-talk-bounces@nzip.org.nz
[mailto:phys-teach-talk-bounces@nzip.org.nz] <b><span style='font-weight:bold'>On
Behalf Of </span></b>Phil Butler<br>
<b><span style='font-weight:bold'>Sent:</span></b> Tuesday, April 08, 2008 9:43
AM<br>
<b><span style='font-weight:bold'>To:</span></b> Rob Campbell; Terry Moffat;
phys-teach-talk@nzip.org.nz<br>
<b><span style='font-weight:bold'>Subject:</span></b> RE: eliptical orbits</span></font><o:p></o:p></p>
</div>
<p class=MsoNormal><font size=3 face="Times New Roman"><span style='font-size:
12.0pt'><o:p> </o:p></span></font></p>
<p class=MsoPlainText><font size=2 face="Courier New"><span lang=EN-AU
style='font-size:10.0pt'>Hi all<o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face="Courier New"><span lang=EN-AU
style='font-size:10.0pt'> <o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face="Courier New"><span lang=EN-AU
style='font-size:10.0pt'>One of the nicest hand waving explanations is <st1:City
w:st="on"><st1:place w:st="on">Newton</st1:place></st1:City>'s own figure which
I have attached. I seem to recall that this email list doesn't take diagrams so
it may be stripped off :-( <o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face="Courier New"><span lang=EN-AU
style='font-size:10.0pt'> <o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face="Courier New"><span lang=EN-AU
style='font-size:10.0pt'>The essence is to imagine standing on a tall hill (or
the <st1:place w:st="on"><st1:PlaceName w:st="on">Auckland</st1:PlaceName> <st1:PlaceName
w:st="on">Sky</st1:PlaceName> <st1:PlaceType w:st="on">Tower</st1:PlaceType></st1:place>)
and throwing a ball harder and harder horizontally.<o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face="Courier New"><span lang=EN-AU
style='font-size:10.0pt'> <o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face="Courier New"><span lang=EN-AU
style='font-size:10.0pt'>Ignore air resistance.<o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face="Courier New"><span lang=EN-AU
style='font-size:10.0pt'>The first ball might hit the ground (earth) 100m away.
<o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face="Courier New"><span lang=EN-AU
style='font-size:10.0pt'>The second ball might hit the ground 100km away. <o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face="Courier New"><span lang=EN-AU
style='font-size:10.0pt'>The third ball might hit the ground 1000km away, well
over the horizon and where the curvature of the earth is important. <o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face="Courier New"><span lang=EN-AU
style='font-size:10.0pt'>The fourth ball, if you throw it with exactly the
right velocity, will just miss the earth all round, and come and hit you on the
back of the head. It will have gone in a circular orbit.<o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face="Courier New"><span lang=EN-AU
style='font-size:10.0pt'>The fifth ball, with an even harder throw, goes in
elliptical orbit, with you as the closest point.<o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face="Courier New"><span lang=EN-AU
style='font-size:10.0pt'> <o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face="Courier New"><span lang=EN-AU
style='font-size:10.0pt'>Note finally that if the earth was a small diameter,
or if your tower or mountain was very high, that balls 1, 2 and 3, would travel
in elliptical orbits with you at their furthest point.<o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face="Courier New"><span lang=EN-AU
style='font-size:10.0pt'> <o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face="Courier New"><span lang=EN-AU
style='font-size:10.0pt'>Phil<o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face="Courier New"><span lang=EN-AU
style='font-size:10.0pt'> <o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face="Courier New"><span lang=EN-AU
style='font-size:10.0pt'><img width=608 height=924 id="_x0000_i1025"
src="cid:image012.gif@01C899CD.1C02A560"><o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face="Courier New"><span lang=EN-AU
style='font-size:10.0pt'> <o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face="Courier New"><span lang=EN-AU
style='font-size:10.0pt'> <o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face="Courier New"><span lang=EN-AU
style='font-size:10.0pt'> <o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face="Courier New"><span lang=EN-AU
style='font-size:10.0pt'> <o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face="Courier New"><span lang=EN-AU
style='font-size:10.0pt'> <o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face="Courier New"><span style='font-size:
10.0pt'>-----Original Message-----<br>
From: phys-teach-talk-bounces@nzip.org.nz
[mailto:phys-teach-talk-bounces@nzip.org.nz] On Behalf Of Rob Campbell<br>
Sent: Monday, 7 April 2008 10:34 p.m.<br>
To: 'Terry Moffat'; phys-teach-talk@nzip.org.nz<br>
Subject: RE: eliptical orbits</span></font><span lang=EN-AU><o:p></o:p></span></p>
<p class=MsoPlainText><font size=2 face="Courier New"><span lang=EN-AU
style='font-size:10.0pt'> <o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face="Courier New"><span lang=EN-AU
style='font-size:10.0pt'>Hi!<o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face="Courier New"><span lang=EN-AU
style='font-size:10.0pt'> <o:p></o:p></span></font></p>
<p class=MsoPlainText><st1:City w:st="on"><st1:place w:st="on"><font size=2
face="Courier New"><span lang=EN-AU style='font-size:10.0pt'>Newton</span></font></st1:place></st1:City><span
lang=EN-AU>'s explanation remains the simplest one that fully explains this,
but<o:p></o:p></span></p>
<p class=MsoPlainText><font size=2 face="Courier New"><span lang=EN-AU
style='font-size:10.0pt'>I won't attempt maths in a text-only email!<o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face="Courier New"><span lang=EN-AU
style='font-size:10.0pt'> <o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face="Courier New"><span lang=EN-AU
style='font-size:10.0pt'>So, here's the hand-waving version.<o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face="Courier New"><span lang=EN-AU
style='font-size:10.0pt'> <o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face="Courier New"><span lang=EN-AU
style='font-size:10.0pt'>Earth satellites like the Moon or the ISS 'fall'
toward Earth. They never<o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face="Courier New"><span lang=EN-AU
style='font-size:10.0pt'>arrive because there's a sideways component to their
velocity. If the ISS<o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face="Courier New"><span lang=EN-AU
style='font-size:10.0pt'>really does move toward Earth, it loses PE and gains
sideways KE. The<o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face="Courier New"><span lang=EN-AU
style='font-size:10.0pt'>sideways motion tends to draw it tangentially away
from Earth - which<o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face="Courier New"><span lang=EN-AU
style='font-size:10.0pt'>doesn't prevent it coming closer (it does.)<o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face="Courier New"><span lang=EN-AU
style='font-size:10.0pt'> <o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face="Courier New"><span lang=EN-AU
style='font-size:10.0pt'>As it approaches Earth, tangential velocity builds up,
slowing then<o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face="Courier New"><span lang=EN-AU
style='font-size:10.0pt'>reversing the Space Station's fall. It starts to
rise, gaining PE as it<o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face="Courier New"><span lang=EN-AU
style='font-size:10.0pt'>does so, and losing KE. At its furthest point
from Earth, its orbital<o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face="Courier New"><span lang=EN-AU
style='font-size:10.0pt'>velocity is lowest; at the nearest point, its orbital
velocity is highest.<o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face="Courier New"><span lang=EN-AU
style='font-size:10.0pt'>This generates an ellipse, though at 10pm I can't
illustrate this in words!<o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face="Courier New"><span lang=EN-AU
style='font-size:10.0pt'>A moment's thought will show that an ellipse is
reasonable; 1/2 hour's<o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face="Courier New"><span lang=EN-AU
style='font-size:10.0pt'>thought with pen and paper will prove it.<o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face="Courier New"><span lang=EN-AU
style='font-size:10.0pt'> <o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face="Courier New"><span lang=EN-AU
style='font-size:10.0pt'>There's no difference in principle between a satellite
orbiting Earth and a<o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face="Courier New"><span lang=EN-AU
style='font-size:10.0pt'>planet orbiting the Sun - just one of scale.
Planets tend to follow<o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face="Courier New"><span lang=EN-AU
style='font-size:10.0pt'>elliptical orbits approximating to circles, maybe on
account of their fiery<o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face="Courier New"><span lang=EN-AU
style='font-size:10.0pt'>origins.<o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face="Courier New"><span lang=EN-AU
style='font-size:10.0pt'> <o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face="Courier New"><span lang=EN-AU
style='font-size:10.0pt'>An ellipse is a squashed circle (apologies,
mathematicians) with a major and<o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face="Courier New"><span lang=EN-AU
style='font-size:10.0pt'>minor axis rather than a constant radius.
Really, a circle is the special<o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face="Courier New"><span lang=EN-AU
style='font-size:10.0pt'>case ellipse where the two axes are equal.<o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face="Courier New"><span lang=EN-AU
style='font-size:10.0pt'> <o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face="Courier New"><span lang=EN-AU
style='font-size:10.0pt'>Hope that helps,<o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face="Courier New"><span lang=EN-AU
style='font-size:10.0pt'> <o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face="Courier New"><span lang=EN-AU
style='font-size:10.0pt'>Rob Campbell<o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face="Courier New"><span lang=EN-AU
style='font-size:10.0pt'> <o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face="Courier New"><span lang=EN-AU
style='font-size:10.0pt'> <o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face="Courier New"><span lang=EN-AU
style='font-size:10.0pt'>-----Original Message-----<o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face="Courier New"><span lang=EN-AU
style='font-size:10.0pt'>From: phys-teach-talk-bounces@nzip.org.nz<o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face="Courier New"><span lang=EN-AU
style='font-size:10.0pt'>[mailto:phys-teach-talk-bounces@nzip.org.nz] On Behalf
Of Terry Moffat<o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face="Courier New"><span lang=EN-AU
style='font-size:10.0pt'>Sent: Monday, 7 April 2008 8:22 a.m.<o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face="Courier New"><span lang=EN-AU
style='font-size:10.0pt'>To: phys-teach-talk@nzip.org.nz<o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face="Courier New"><span lang=EN-AU
style='font-size:10.0pt'>Subject: eliptical orbits<o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face="Courier New"><span lang=EN-AU
style='font-size:10.0pt'> <o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face="Courier New"><span lang=EN-AU
style='font-size:10.0pt'>Hi,<o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face="Courier New"><span lang=EN-AU
style='font-size:10.0pt'> <o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face="Courier New"><span lang=EN-AU
style='font-size:10.0pt'>can someone give me a simple reason why planets move
in eliptical orbits.<o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face="Courier New"><span lang=EN-AU
style='font-size:10.0pt'> <o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face="Courier New"><span lang=EN-AU
style='font-size:10.0pt'> <o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face="Courier New"><span lang=EN-AU
style='font-size:10.0pt'>does it have something to do with the fact that the
system is moving through<o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face="Courier New"><span lang=EN-AU
style='font-size:10.0pt'>space?<o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face="Courier New"><span lang=EN-AU
style='font-size:10.0pt'> <o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face="Courier New"><span lang=EN-AU
style='font-size:10.0pt'> <o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face="Courier New"><span lang=EN-AU
style='font-size:10.0pt'>ta Terry<o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face="Courier New"><span lang=EN-AU
style='font-size:10.0pt'> <o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face="Courier New"><span lang=EN-AU
style='font-size:10.0pt'> <o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face="Courier New"><span lang=EN-AU
style='font-size:10.0pt'> <o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face="Courier New"><span lang=EN-AU
style='font-size:10.0pt'> <o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face="Courier New"><span lang=EN-AU
style='font-size:10.0pt'>_______________________________________________<o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face="Courier New"><span lang=EN-AU
style='font-size:10.0pt'>Phys-teach-talk mailing list<o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face="Courier New"><span lang=EN-AU
style='font-size:10.0pt'>Phys-teach-talk@nzip.org.nz<o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face="Courier New"><span lang=EN-AU
style='font-size:10.0pt'>http://nzip.org.nz/mailman/listinfo/phys-teach-talk_nzip.org.nz<o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face="Courier New"><span lang=EN-AU
style='font-size:10.0pt'> <o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face="Courier New"><span lang=EN-AU
style='font-size:10.0pt'> <o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face="Courier New"><span lang=EN-AU
style='font-size:10.0pt'>_______________________________________________<o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face="Courier New"><span lang=EN-AU
style='font-size:10.0pt'>Phys-teach-talk mailing list<o:p></o:p></span></font></p>
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style='font-size:10.0pt'>Phys-teach-talk@nzip.org.nz<o:p></o:p></span></font></p>
<p class=MsoPlainText><font size=2 face="Courier New"><span lang=EN-AU
style='font-size:10.0pt'>http://nzip.org.nz/mailman/listinfo/phys-teach-talk_nzip.org.nz<o:p></o:p></span></font></p>
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